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Lp stability of networked control systems implemented
on WirelessHART
Alejandro I. Maass, Dragan Nešić, Romain Postoyan, Peter Dower
To cite this version:
Alejandro I. Maass, Dragan Nešić, Romain Postoyan, Peter Dower.
Lp stability of networked
control systems implemented on WirelessHART. Automatica, Elsevier, 2019, 109, pp.108514.
�10.1016/j.automatica.2019.108514�. �hal-02268982�
L
p
stability of networked control systems implemented on
WirelessHART ?
Alejandro I. Maass
a, Dragan Neˇsi´
c
a, Romain Postoyan
b, Peter M. Dower
aa
Department of Electrical and Electronic Engineering, The University of Melbourne, Parkville, 3010, Victoria, Australia
b
Universit´e de Lorraine, CNRS, CRAN, F-54000 Nancy, France
Abstract
This paper provides results on input-output Lpstability of networked control systems (NCSs) implemented over WirelessHART
(WH). WH is a communication protocol widely used in process instrumentation. It is mainly characterised by its multi-hop structure, slotted communication cycles, and the possibility to simultaneously transmit over different frequencies. We propose a non-linear hybrid model of WH–NCSs that is able to capture these network functionalities, and that it is more general than existing models in the literature. Particularly, the multi-hop nature of the network is translated into an interesting mathematical structure in our model. We then follow the emulation approach to stabilise the NCS. We first assume that we know a stabilising controller for the plant without the network. We subsequently show that, under reasonable assumptions on the scheduling protocol, stability is preserved when the controller is implemented over the network with sufficiently frequent data transmission. Specifically, we provide bounds on the maximum allowable transmission interval (MATI) under which all protocols that satisfy the property of being persistently exciting (PE) lead to Lpstable WH–NCSs. These bounds exploit the
mathematical structure of our WH–NCS model, improving the existing bounds in the literature. Additionally, we explain how to schedule transmissions over the hops to satisfy the PE property. In particular, we show how simultaneous transmissions over different frequency channels can be exploited to further enlarge the MATI bound.
Key words: Networked control, WirelessHART, Non-linear systems, Lpstability.
1 Introduction
Networked control systems (NCSs) have received much interest during the last years due to their many practi-cal implications [11, 13]. While the available results are able to capture the essential effects that communication constraints have on the closed-loop system, it remains unclear how these results can be applied to specific phys-ical networks. In this paper, we are motivated to de-velop results tailored to NCSs implemented over wire-less multi-hop networks, which are increasingly used in industry control. Particularly, we study WirelessHART (WH), the first open wireless communication standard for measurement and control in the process industries [1].
? Corresponding author A.I. Maass. This work was sup-ported by the ARC Discovery Scheme, grant number DP170104099, and by the ANR project COMPACS, ANR-13-BS03-004-02.
Email addresses: [email protected] (Alejandro I. Maass), [email protected] (Dragan Neˇsi´c), [email protected] (Romain Postoyan), [email protected] (Peter M. Dower).
According to many international companies, it serves as a compelling alternative for industrial control, as it improves operations, increases productivity, and saves money [1]. WH is a mesh network which utilises field de-vices in a multi-hop fashion and schedule these via time division multiple access (TDMA).
WH has received a lot of attention in the last decade, both heuristically [4, 8, 10, 21] and analytically [2, 3, 7, 20, 28, 29, 31, 32]. In particular, scheduling in WH is ad-dressed in [20, 28, 29, 31], where several scheduling pro-tocols tailored to different networking requirements can be found. These works focus on the network itself and do not use it in a control loop. Regarding control sys-tems that use a WH network to transmit packets in the control loop, we can find controller-communication co-design in [7, 32]. The authors in [32] developed an LQG framework to study minimum-energy packet forwarding policies for communicating sensor measurements over a WH network. The recent work [7] completes the liter-ature by considering both packet-forwarding and con-troller co-design. Modelling of WH–NCSs is addressed in [2, 3]. In [3], the authors propose a mathematical
frame-work for modelling and analysis of multi-hop netframe-works designed for linear discrete-time systems consisting of multiple control loops closed over a multi-hop wireless communication network. They separate control, topol-ogy, routing, and scheduling and propose formal syntax and semantics for the dynamics of the composed sys-tem, providing an explicit translation of multi-hop con-trol networks to switched systems. A similar work is [2], in which the formulation, modelling and design of a fixed structure topology for potential application in wireless NCSs is provided. The setup consists of a discrete-time plant, an output feedback discrete-time controller, and intermediate transfer and receiving networks.
The first purpose of this study is to provide a mod-elling framework that encompasses the models used in [2,3,7,32] by considering a more general class of mod-els that result from a careful study of the WH network. Note that the existing models of WH–NCSs in [2,3,7,32] assume the plant and controller to be discrete-time lin-ear systems. That is, the results are valid only at each transmission instant, but the inter-sample behaviour is lost. This also implies that transmissions in the WH network happen equidistantly. Such assumptions may be hard to satisfy in WH, where extra features need to be considered, e.g. possibly non-linear plant and controller, field-device dynamics, time-varying trans-mission instants, inter-transtrans-mission behaviour given by the continuous plant dynamics, and at-transmission behaviour given by packet transmission. We propose a hybrid model of WH–NCSs that is able to capture all the latter features. More importantly, the multi-hop nature of the network is revealed through our model via an interesting mathematical structure which turns out to be key in our stability analysis.
An important part of the model is the so-called protocol equation, which defines how field devices are scheduled in the superframe table of a WH network. It is desired that scheduling protocols satisfy certain properties in or-der to state our results. A key property is the so-called persistence of excitation in T (P ET) [24]. This
prop-erty was presented in [24] and requires the existence of a fixed number of transmissions T within which all net-work nodes are visited by the protocol. In WH netnet-works, this translates into every field device being scheduled to transmit within T transmissions. We present a large class of scheduling protocols for which the P ETproperty
holds. Specifically, we provide three relevant examples that belong to this class and that exploit the flexibility of WH networks, i.e. multiple transmissions over differ-ent frequency channels. These results can also be used to design other scheduling policies for field devices in WH.
It was shown in [24] that P ET protocols lead to Lp
sta-ble NCSs under extra assumptions. However, these re-sults are derived for generic non-linear NCSs in which a packet is immediately received by the controller/plant whenever nodes have access to the network. Therefore,
the potential structure of a physical network is not ex-ploited, e.g. multi-hop in WH. We go one step further to [24] and study Lpstability in WH–NCSs, for which we
exploit the mathematical structure of our hybrid model. Specifically, we analyse Lp stability in the case where
the controller is designed by emulation. The main idea of emulation is to first design a controller that stabilises the plant in the absence of the network. Then, the controller is implemented over the network and it is shown that the Lp stability of the system is preserved, see e.g. [26]. In
particular, stability is preserved if the scheduling proto-cols are persistently exciting, and if data is transmitted at a high enough rate, measured by the maximum al-lowable transmission interval (MATI). We provide two different and easily computable MATI bounds and il-lustrate in examples that our MATI bounds result to be significantly larger than the ones in [24]. In our pre-liminary work [16, 17], we presented MATI bounds for disturbance-free WH–NCSs, which resulted to be quite conservative. With this article, we extend our previous results in [16, 17] by considering external disturbances and by providing much tighter MATI bounds.
In brief, the primary contributions of this paper are:
(1) We provide a hybrid model for WH–NCSs that cap-tures both at- and inter-transmission behaviour, time-varying transmission instants, field device dy-namics, and non-linear plant and controller, thus generalising the models in [2, 3, 7, 32].
(2) Our model reveals a mathematical structure that comes directly from the features of WH, i.e. multi-hop and buffer dynamics. Previous generic models for non-linear NCSs like [5,18,24,26], although they can be used to analyse our WH–NCS, these do not exploit these features and thus provide more con-servative results.
(3) We provide a large class of protocols that satisfy the property of being P ET, which is natural in WH
networks. We provide three relevant examples that are implementable in WH and belong to this class. (4) We provide two different and easily computable MATI bounds that ensure the Lp stability of our
WH–NCS, and that exploit the mathematical structure of our model. We illustrate in examples that our MATI bounds result to be significantly larger than the ones in [24].
(5) We extend our previous results in [16,17] by consid-ering external disturbances and by providing sig-nificantly less conservative MATI bounds.
The paper is organized as follows. Notation and prelimi-naries are given in Section 2. Section 3 describes the WH standard in detail. We present the WH–NCS model in Section 4 and scheduling is analysed in Section 5. Lp
sta-bility results are stated in Section 6. A linear case study is addressed in Section 7. Numerical examples are given in Section 8, whilst Section 9 draws conclusions.
2 Preliminaries
2.1 Notation
Denote by R the set of real numbers, Rn the set of all real vectors with n components, and Rm×n the set of
all real matrices of dimension m × n. Let R≥0= [0, ∞),.
Z≥0 = {0, 1, 2, . . . }, and N. = {1, 2, 3, . . . }. Let A. n≥0
denote the set of all n × n matrices with nonnegative entries, and let Rn
≥0 denote the nonnegative orthant of
Rn. A function α : R≥0 → R≥0 is of class K if it is
continuous, zero at zero and strictly increasing. It is of class K∞ if it is of class K and unbounded. A function
β : R≥0× R≥0 → R≥0is of class KL if β(·, t) is of class K
for each t ≥ 0, and if β(s, ·) is continuous, non-increasing and satisfies limt→∞β(s, t) = 0 for each s ≥ 0. Given
t ∈ R and a piecewise continuous function f : R → Rn,
we use the notation f (t+) = lim.
s→t,s>tf (s). For
sim-plicity, we use (x, y) = [x. T yT]T ∈ Rn+m, for any
x ∈ Rn
and y ∈ Rm. For a vector x = (x
1, . . . , xn) ∈ Rn, kxkp . = (Pn i=1|xi|p) 1/p
, for all p ∈ [1, ∞), and kxk∞ =.
maxi|xi|. To ease notation, we use |x| to denote kxk2.
The same notation is used to denote the induced 2-norm of a matrix. For an m × n matrix A, the induced ma-trix 1-norm is given by kAk1
.
= max1≤j≤nP m i=1|aij|.
In stands for the n × n identity matrix. We also use
IN n
.
= [In · · · In]N to denote the matrix [In · · · In] ∈
Rn×N n. We will often consider vectors of the form ¯x, where x ∈ Rn and ¯x= (|x.
1|, . . . , |xn|)T. For a function
f : R → Rn, we define ¯f : t → f (t). Let Df (t) denote the left-handed derivative of f : R → Rn, if it exists,
i.e. Df (t)= lim. h→0,h<0
f (t+h)−f (t)
h . We define 1Sas the
function 1S : N → {0, 1} such that 1S(i) = 1 if i ∈ S,
and 1S(i) = 0 if i /∈ S. Let f : R → Rn be a (Lebesgue)
measurable function and define
kf kLp=. Z R kf (s)kpds 1/p , (1)
for p ∈ N, kf kL∞= ess sup. t∈Rkf (t)k, and kf kL∞[a,b] =.
ess supt∈[a,b]kf (t)k. Note that k · k in (1), for any p ∈ [1, ∞], can be any p-norm on Rn [14]. We use the
Eu-clidean norm k · k = | · | as default throughout the pa-. per. However, at the end of Section 6 we use the p-norm on Rn, i.e. k · k= k · k. p, which will become clear
from the context. Let f : R → Rn
and let [a, b] ⊂ R, we use the notation kf kLp[a,b] =. R[a,b]kf (s)k
pds1/p , to denote the Lp norm of f when restricted to the
in-terval [a, b]. For T ∈ N, we define a collection of se-quences of n×n matrices Sn(T ), such that, for all s ∈ N,
{Ai}i∈N ∈ Sn(T ) if and only if Q s+T −1
i=s Ai = 0. Let
x = (x1, . . . , xn), y = (y1, . . . , yn) ∈ Rn. A partial order
is given by x y ⇐⇒ xi≤ yi, for all i ∈ {1, . . . , n}.
We define an analogous partial order on elements of An ≥0
in the natural way, i.e. A B ⇐⇒ B − A ∈ An ≥0.
2.2 Underlying stability theory
Consider the jump-flow (hybrid system) model Σ,
˙
z = f (t, z, w), t ∈ [ti, ti+1], (2a)
z(t+i ) = h(i, z(ti)), (2b)
y = H(t, z), (2c)
where z ∈ Rnzis the state, w ∈ Rnwis an exogenous
per-turbation, y ∈ Rny is a prescribed output, n
z, nw, ny ∈
N, and {ti}∞i=0 is a sequence of increasing time instants
such that, for some τ ∈ R and ε > 0, ε < ti+1− ti< τ <
∞ for all i ∈ N. Suppose Σ is initialised at (t0, z0) with
input w. We assume enough regularity on f and h to guarantee existence of the solution z(·) = z(·, t0, z0, w)
on the interval of interest, see [18].
We now define the stability notions used throughout this paper.
Definition 1 Let p ∈ N∪{+∞} and γ ≥ 0 be given. We say that Σ is Lp stable from w to y with gain γ if there
exists K ≥ 0 such that kykLp[to,t] ≤ K|z0| + γkwkLp[t0,t],
for all t ≥ t0, w ∈ Lp[t0, t] and z0∈ Rnz.
Definition 2 Let p, q ∈ N ∪ {+∞} and γ ≥ 0 be given. The state z of Σ is said to be Lp to Lq detectable
from (y, w) with gain γ if there exists K ≥ 0 such that kzkLq[t0,t] ≤ K|z0| + γkykLp[t0,t]+ γkwkLp[t0,t], for all
t ≥ t0, y ∈ Lp[t0, t], w ∈ Lp[t0, t] and z0∈ Rnz.
Consider the feedback interconnection of two systems Σ1and Σ2, each one of the same form as Σ in (2), that is
Σ1 ˙ x1 = f1(t, x1, x2, w), t ∈ [ti, ti+1], x1(t+i ) = h1(i, x1(ti)), y1 = H1(t, x1, y2, w), (3) Σ2 ˙ x2 = f2(t, x1, x2, w), t ∈ [ti, ti+1], x2(t+i ) = h2(i, x2(ti)), y2 = H2(t, y1, x2, w). (4)
This interconnection admits a small-gain theorem pre-sented in [18, Section II-B], which will later be used to prove our main stability result.
Theorem 1 Suppose that p, q ∈ N ∪ {+∞}, and the following hold:
(1) System (3) is Lpstable from (y2, w) to y1with gain
γ1;
(2) x1in (3) is Lpto Lq detectable from (y1, w);
(3) (4) is Lpstable from (y1, w) to y2with gain γ2;
(4) x2in (4) is Lpto Lq detectable from (y2, w);
(5) The small-gain condition γ1γ2< 1 holds.
Plant Controller WirelessHART Field Devices Handheld device Network Manager Gateway
Fig. 1. WirelessHART architecture.
We use the standard notion of uniform global exponen-tial stability (UGES) for system (3)-(4) in the absence of exogenous perturbations.
Definition 3 Consider system Σ and suppose that w =. 0. We say that the origin of Σ is uniformly globally ex-ponentially stable if there exist K, L ≥ 0 such that, for every z0∈ Rnz, |z(t, t0, z0)| ≤ K exp(−L(t − t0))|z0| for
all t ≥ t0.
The following sufficient condition is used to ensure UGES of (3)-(4) in the absence of w [24].
Theorem 2 Suppose that systems (3) and (4) sat-isfy all hypotheses of Theorem 1. If there exist L1, L2, L3, L4≥ 0 such that |f1(t, x1, x2, 0)| ≤ L1(|x1| +
|x2|), |f2(t, x1, x2, 0)| ≤ L2(|x1| + |x2|), |h1(i, x1)| ≤
L3|x1|, |h2(i, x2)| ≤ L4|x2|, for all x1 ∈ Rnx1, x2 ∈
Rnx2, nx
1, nx2 ∈ N, all t ≥ t0and all i ∈ N. Then,
(3)-(4) with w= 0 is UGES.. 3 WirelessHART Network
In the following, a description of WH and the adopted assumptions are provided.
3.1 Communication features
The general architecture of WH is shown in Fig. 1. It con-sists of an interconnection of basic components including field devices (sensor, actuators and routers), handheld devices, gateways, and a network manager. WH is based on the IEEE 802.15.4-2006 physical layer and operates in the 2.4 GHz ISM radio band with a maximum data rate of 250 kbps over 15 frequency division multiplexed chan-nels. In the data link layer, WH defines a slotted TDMA technology. That is, each frequency channel is subdi-vided into timeslots in which an assigned field device is allowed to transmit. WH networks also support multi-ple access timeslots, where multimulti-ple devices can share a specific channel. In this case, WH uses carrier sense mul-tiple access with collision avoidance (CSMA/CA) mech-anisms to avoid collisions. We restrict our attention to TDMA in this paper. We study WH–NCSs under CSMA in our recent work [15].
Fig. 2. WirelessHART superframe table.
3.2 TDMA superframe structure
All communications in a WH network are defined with respect to a superframe. For each l-th channel, l = 1, . . . , 15, a superframe is an a priori fixed period of time Tl > 0, contiguous in real time with other
su-perframes within each channel, that is divided into a sequence of timeslots as depicted in Fig. 2. Field devices are scheduled to transmit in the superframe, and each one of the 15 channels may have a different superframe depending on the chosen scheduling protocol. The set of superframes across frequency channels is called a superframe table. Each timeslot is strictly Ts = 10[ms]
in duration. Within this timeslot, a complete single data packet and its corresponding acknowledgement are transmitted between two field devices. The transmis-sion delay required for the delivery of a packet from a transmitting device to a receiving device, in each l-th channel and i-th timeslot, is denoted by tRX
l,i − tTXl,i and
it depends on the packet size. For each channel l-th channel and at the end of every i-th timeslot, the time it takes to acknowledge such a packet is denoted by τACK
l,i . For effective TDMA communications, all devices
need to be synchronized. This is ensured, at the begin-ning of each i-th timeslot on the l-th channel, by the amount τSYNC
l,i . For simplicity, the following additional
assumption is adopted.
Assumption 4 The following holds.
(a) Transmissions across all channels are synchro-nized, i.e. τSYNC
i
.
= τSYNC
l,i = τk,iSYNC for all
l, k ∈ {1, . . . , 15}. We define ε = inf. i∈NτSYNC
i and
assume that ε > 0.
(b) Acknowledgement time is negligible in each timeslot, i.e. τACK
l,i = 0 for all l ∈ {1, . . . , 15} and i ∈ N.
(c) Packets are transmitted instantaneously in every timeslot, i.e. ti = t. TXl,i = tRXl,i for all l ∈ {1, . . . , 15}
and i ∈ N. We refer to ti as transmission instant.
(d) One successful transmission between devices occurs within each timeslot, per channel.
Item (a) is adopted to avoid accounting for clock drift. This assumption is reasonable under the strict synchro-nization procedure in WH [1]. However, this assumption can be relaxed, and the analytical tools we present in this paper can be used to deal with such scenario. Notation in that case becomes cumbersome and we thus adopt Assumption 4(a) for clarity. In Section 8, we present a simulation in which the WH network is subject to clock drift, and we illustrate that our results still work un-der this constraint. Items (b) and (c) are reasonable in the context of WH in practice. Note that each times-lot is strictly 10[ms], and the WH specification allocates 832[µs] for the whole ACK packet. The timeslot length is about 12 times the ACK packet, hence why neglecting it is appropriate. With respect to item (c), please note that the maximum packet length in WH is 127 bytes [6]. We allude to papers like [8, 21], in which the authors have implemented the NCS over a real WH network and measured packet transmission delays, among other val-ues. In these experiments, we can see that the transmis-sion delay ranges from 0.919[ms] to 1[ms]. The timeslot length is then about 10 times the transmission delay, hence why we believe it is appropriate to neglect it. We adopt item (d) so as to restrict attention to the effects of the transmission instants tiin the modelling. Packet
drops are outside the scope of this paper, but are anal-ysed in our recent paper [15] with an analogue stochastic framework.
Remark 5 The transmission instants ti need not be
equally spaced between each other depending on synchro-nization time τSYNC
i . This models the fact that real-life
networks in general have time-varying transmission in-stants and not necessarily equidistant as often modelled,
see [2, 3, 7, 32].
We will parametrize our model with the so-called maxi-mal allowable transmission interval (MATI) [26], which we denote as τ > 0. It is a measure of how fast the net-work needs to transmit in order to preserve stability of the NCS. The next corollary comes from Assumption 4.
Corollary 1 If Assumption 4 holds, the transmission instants ti in the WH network, satisfy ε ≤ ti+1− ti ≤
τ ≤ 2Ts− ε for all i ∈ N.
Corollary 1 states that a packet must be transmitted at most in τ seconds, which cannot be larger than 2Ts− ε.
4 Model of a WH–NCS
In this section we provide the hybrid modelling frame-work for WH–NCSs. Consider Fig. 3, where the plant is modelled as a general non-linear system via
˙
xp= fp(xp, ˆu, w), y = gp(xp), (5)
Fig. 3. Block diagram of a NCS over WirelessHART.
Fig. 4. NCS closed over a WH network with `yfield devices
in the y-path and `ufield devices in the u-path.
where xp∈ Rnpis the state, ˆu ∈ Rnuis the control signal
received by the plant, y ∈ Rny is the plant output, w ∈
Rnw is an exogenous perturbation which is assumed to belong to Lp, i.e. its Lpnorm is finite for given p ∈ [1, ∞],
and np, nu, ny, nw ∈ N. The controller is also modelled
as a non-linear system given by
˙
xc= fc(xc, ˆy, w), u = gc(xc), (6)
where xc∈ Rnc is the state of the controller, ˆy ∈ Rny is
the plant output received by the controller, u ∈ Rnu is
the control signal, and nc ∈ N. The functions fp, fc are
assumed to be continuous and gp, gc are assumed to be
continuously differentiable.
We now model the WH network based on the descrip-tion and assumpdescrip-tions provided in Secdescrip-tion 3. Particularly, we model the WH network as in Fig. 4, i.e. we consider `y ∈ Z≥0 field devices interconnected in the
plant-to-controller path (y-path), and `u∈ Z≥0in the
controller-to-plant path (u-path). We label field devices as Dy αand
Du
β, where α = 1, . . . , `yand β = 1, . . . , `u. For each field
device, its inputs and outputs are depicted in Fig. 4. The signal that reaches the controller (resp. plant) in Fig. 4, i.e. y`y (resp. u`u), is denoted as ˆy (resp. ˆu) to be
consis-tent with Fig. 3 and exisconsis-tent NCS literature. Each field device acts as a router for data from/to neighbouring field devices and we model them as buffers. We introduce buffer state variables by
α and buβ for field devices in the
y-path and u-path, respectively. We now explain the re-ception and transmission behaviour of field devices, and we present the corresponding equations. We treat field devices as zero-order-hold devices, in which their buffer value and output are held between transmissions. Reception: Suppose a field device Dy
αreceives a packet
at time instant ti. Then, Dαy updates the content of its
buffer via its input. During this process, the output of Dαy remains unchanged. We write this as follows,
˙
yα(t) = 0, ˙byα(t) = 0, t ∈ [ti, ti+1], (7a)
for all α = 1, . . . , `y. Note that y0
.
= y for α = 1, i.e. device one samples the value of the plant output. Transmission: Suppose a field device Dy
α is scheduled
to transmit at time instant ti. Here, Dαysends the content
of its buffer through its output, and keeps it until a new packet is received. This can be written as follows,
˙ yα(t) = 0, ˙byα(t) = 0, t ∈ [ti, ti+1], (8a) yα(t+i ) = b y α(ti), byα(t + i) = b y α(ti), (8b) for all α = 1, . . . , `y.
Motivated by (8)-(7), we introduce the network-induced error ζ ∈ Rnζ, with n
ζ
.
= nζy + nζu, nζy = 2`. yny,
and nζu = 2`. unu. We define it as ζ = (ζ. y, ζu), where
ζy ∈ Rnζy and ζu ∈ Rnζu are the corresponding errors
of the y-path and u-path, respectively, and are given by
ζy= ζ1y, ζ2y, . . . , ζ`y y, ζ y `y+1, ζ y `y+2, . . . , ζ y 2`y . = by1− y, by2− y1, . . . , by`y − y`y−1, y1− by1, y2− by2, . . . , y`y− b y `y, (9a) ζu= ζ1u, ζ2u, . . . , ζ`u u, ζ u `u+1, ζ u `u+2, . . . , ζ u 2`u . = bu1− u, bu 2− u1, . . . , bu`u− u`u−1, u1− bu1, u2− bu2, . . . , u`u− b u `u. (9b)
The first `?components of ζ?, ? ∈ {y, u}, are related to
the buffer update during reception, and we call these re-ception errors. The remaining `? components of ζ? are
related to the transmission of such buffer value through their output, and we call them transmission errors. In particular, we reset to zero these errors to model recep-tion and transmission. This is a major difference with previous models of non-linear NCSs like [5, 18, 26], in which the network-induced error is given by e = (ˆ. y − y, ˆu − u) (i.e. no specific network is considered, and the possible buffer dynamics are ignored). Specifically, whenever there is a transmission, this error models it as the plant (resp. controller) receiving the sensor (resp. ac-tuator) packet immediately. In WH networks, this is not always the case, as the packet needs to travel along dif-ferent field devices before it reaches its destination, gen-erating an intrinsic delay. Therefore, dynamics of field devices have to be taken into account, together with a proper definition of the network-induced error, as in (9).
Now that all components of Fig. 4 have been modelled, we are in a position to present the model for a WH– NCS. Define the augmented state x = (x. p, xc), where
x ∈ Rnx. By using (5), (6), (7), (8), and (9), we present
a hybrid model for the block diagram in Fig. 4,
˙ x(t) = f (x(t), ζ(t), w), t ∈ [ti, ti+1], (10a) ˙ ζ(t) = g(x(t), ζ(t), w), t ∈ [ti, ti+1], (10b) x(t+i ) = x(ti), (10c) ζ(t+i ) = H(i)ζ(ti), (10d) where i ∈ N, f : Rnx × Rnζ × Rnw → Rnx and g : Rnx× Rnζ× Rnw → Rnζ are defined as f (x, ζ, w)=. fp(xp, I2`unu · ζ u+ g c(xc), w), fc(xc, I2`nyy · ζ y+ g p(xp), w) , (11) g(x, ζ, w)=. −∂gp ∂xp fp(xp, I2`nuu· ζ u+ g c(xc), w), 0, . . . , 0, − ∂gc ∂xc fc(xc, I2`yny · ζ y+ g p(xp), w), 0, . . . , 0 , (12)
and where H(i) is a time-varying matrix that decides when field devices are scheduled to transmit. In particu-lar, this matrix will zero components of ζ(t+i ) according to how devices are scheduled in the superframe table. We refer to ζ(t+i ) = H(i)ζ(ti) as the protocol equation,
which is properly studied in Section 5.
5 Scheduling protocols
In this section, we first explain how scheduling protocols can be translated into the form (10d). We then define a large class of protocols for which its protocol equation satisfies the property of being persistently exciting. This property will be key to ensuring Lpstability of (10). To
finalise, we give three relevant protocols which belong to this class and that exploit the flexibility of the network.
5.1 Constructing the protocol equation
In order to have proper TDMA communications in WH, the construction of the superframe table has to satisfy the following requirements from the WH standard [1]:
(i) Only one transmission per frequency channel per timeslot may be scheduled between field devices. (ii) A field device cannot transmit and receive at the
same time.
(iii) All 15 available frequency channels may be used to schedule transmissions.
(iv) Superframes along frequency channels may have different periods.
To construct the protocol equation (10d) associated to the superframe table, we need to follow the above re-quirements together with the dynamics (8)-(7) of the field devices. We proceed to illustrate this process by a simple example. Let us consider Table 1, where devices are scheduled in a WH network with `y = 2, `u = 0,
using a single frequency channel, denoted by CH1. Note
that ζ = ζy given that `u= 0. Based on (8), (7) and
Ta-ble 1, we now show how the network error ζybehaves at
t+i , for i ∈ N. At time instant t1, D y
1 updates its buffer
Table 1
Example of a superframe table constructed under the re-quirements of the WH standard.
t1 t2 t3 CH1 P → D y 1 D y 1 → D y 2 D y 2 → C
buffer values remain constant. Hence, the reception er-ror associated with Dy1 gets reset to zero, that is
ζy(t+1) = by1(t+1) − y(t+1) by2(t+1) − y1(t+1) y1(t+1) − b y 1(t + 1) y2(t+1) − b y 2(t + 1) = 0 ζ2y(t1) y1(t1) − y(t1) ζ4y(t1) = 0 0 0 0 0 Iny 0 0 Iny 0 Iny 0 0 0 0 Iny ζ y(t 1). At time instant t2, D y
1transmits its buffer value towards
Dy2. After this transmission the buffer value by1 remains unchanged, and by2 gets updated. Hence, the transmis-sion error associated with Dy1gets reset to zero, and the reception error associated with Dy2 gets reset to zero,
ζy(t+2) = ζ1y(t2) 0 0 y2(t2) − b1(t2) = Iny 0 0 0 0 0 0 0 0 0 0 0 0 InyInyIny ζ y (t2).
Finally, at time instant t3, D2ytransmits its buffer value
to the controller, thus the transmission error associated with D2ygets reset to zero. That is,
ζy(t+3) = by1(t+3) − y(t+3) by2(t+3) − y1(t+3) y1(t+3) − b y 1(t + 3) y2(t+3) − b y 2(t + 3) = Iny 0 0 0 0 Iny 0 0 0 0 Iny0 0 0 0 0 ζ y(t 3).
Consequently, the above generates the protocol equation ζy(t+
i) = H y(i)ζy(t
i), i ∈ N, where Hy(i) is a
time-varying matrix of the form
Hy(i) = ∆y(i) 0 I − ∆y(i) Γy(i) ,
where ∆y(i) = diagδy
1(i)Iny, δ y 2(i)Iny , and Γy(i) = γ1y(i)Iny 0 (1 − γ1y(i))Iny γ y 2(i)Iny ,
with δyα(i) = 0 when i = α + 3σ and δyα(i) = 1 other-wise, and γy
α(i) = 0 when i = α + 1 + 3σ and γαy(i) = 1
otherwise, for α = 1, 2 and σ ∈ Z≥0. Note that the
su-perframe table is periodic, thus the term 3σ in the above equations. This process can be followed to construct the protocol equation from any superframe table.
In the general case, H(i)= diag {H. y(i), Hu(i)}, where
H?(i)=. ∆?(i) 0 I − ∆?(i) Γ?(i) , (13) ∆?(i)= diag. δ? 1(i)Iny, . . . , δ ? `?(i)Iny , Γ?(i)=. γ? 1(i)Iny (1 − γ? 1(i))Inyγ ? 2(i)Iny . .. . .. (1 − γ? `?−1(i))Inyγ ? `?(i)Iny ,
with ? ∈ {y, u}, and γy
α(i), γβu(i), δ y
α(i), δβu(i) ∈ {0, 1},
α = 1, . . . , `y, β = 1, . . . , `u, which are defined
differ-ently according to the constructed superframe table. Therefore, given a superframe table, we can construct the protocol equation as done for the example above, and the resulting H(i) matrix will have the form in (13), with specific definitions of γy
α(i), γβu(i), δαy(i), δuβ(i) depending
on the chosen table. Later in this section, we provide rel-evant scheduling protocols that arise in WH networks, for which we give these definitions.
5.2 A class of persistently exciting protocols
We would like to implement scheduling protocols that satisfy the property of being persistently exciting. In this section, we provide a large class of TDMA scheduling protocols that satisfy this property.
Definition 6 The protocol (10d) is said to be persis-tently exciting in T (P ET) if there exists T ∈ N such that
i+T −1
Y
k=i
H(k) = 0, (14)
for every i ∈ N.
Recall that transmissions are modelled by resetting to zero components of ζ(t+i ). Then, we can interpret P ET
protocols in WH as protocols that regularly schedule to transmit every field device within a fixed period of time. Motivated by this, we state the following assumption that will allow us to define the class of P ET protocols.
Assumption 7 Every field device Dy
α, Duβ, α =
1, . . . , `y, β = 1, . . . , `u, needs to be scheduled to transmit
within a fixed period of time. Assumption 7 is naturally satisfied by many network technologies such as Ethernet, IEEE 802.11, and IEEE 802.15.4 standard [24], including WirelessHART (WH) networks as we illustrate in Section 5.3. The reason be-hind it is that Assumption 7 only requires that every field device in the path transmits within a fixed period of time. This is not hard to satisfy as the schedule is de-signed by the network manager. Additionally, protocols
that have been designed optimally w.r.t. a certain crite-ria also satisfy Assumption 7, see. e.g. [29, 30].
The next lemma states that, if a scheduling protocol satisfies Assumption 7, then such protocol is P ET. The
proof is provided in the appendix.
Lemma 8 Consider the WH–NCS (10) and suppose As-sumption 7 holds. Then, the corresponding protocol equa-tion (10d) satisfies (14) for some T ∈ N. We are now in a position to define a class of schedul-ing protocols that can be implemented in WH and that satisfy the P ET property.
Definition 9 (Class of WH-P ET protocols) This
class contains all scheduling protocols that satisfy As-sumption 7, and thus satisfy the property of being P ET
according to Lemma 8.
Constructing scheduling protocols that belong to this class is important. In fact, in Section 6 we show that P ET protocols lead to Lpstable WH–NCSs. Therefore,
Assumption 7 can be used to design different types of scheduling policies. We now present three relevant ex-amples that belong to the class of WH-P ET protocols in
Definition 6. These are important to illustrate how the flexibility offered by the WH network can be exploited (multiple frequency channels).
5.3 Examples of P ET scheduling protocols
5.3.1 Simple Round Robin (S-RR)
This protocol schedules the field devices in a round-robin manner [18], i.e. in a predetermined and cyclic manner. A single frequency channel is used and the field devices communicate one after the other. In particular, we adopt the superframe table shown in Table 2.
Table 2
Superframe table for the S-RR protocol.
t1 · · · t`y+1 t`y+2 · · · t`y+`u+2
CH1P → D1y· · · D y `y → C C → D u 1· · · D u `u → P
For this scheduling protocol, and enlightened by Table 2 and the constructive procedure of Section 5.1, it is possible to show that, δαy(i)
. = 1 − 1Sy α(i), δ u β(i) . = 1 − 1Su β(i), γ y α(i) . = 1−1S¯y α(i), and γ u β(i) . = 1−1S¯u β(i), where Sy α . = {i ∈ N : i = α + (`y+ `u+ 2)σ, σ ∈ Z≥0} , Sβy= {i ∈ N : i = β + `. y+ 1 + (`y+ `u+ 2)σ, σ ∈ Z≥0} , ¯ Sy α . = {i ∈ N : i = α + 1 + (`y+ `u+ 2)σ, σ ∈ Z≥0} , ¯ Sβy= {i ∈ N : i = β + `. y+ 2 + (`y+ `u+ 2)σ, σ ∈ Z≥0} ,
for α = 1, . . . , `yand β = 1, . . . , `u. It can be shown that
the parameter T in Lemma 8 is given by T = TS-RR =. max{2`y+ `u+ 2, 2`u+ `y+ 2}.
5.3.2 Frequency Division Duplex Round Robin (FDD-RR)
This scheduling protocol establishes a full-duplex com-munication link that uses two different frequency chan-nels for measurements and actuation operations, see Ta-ble 3. For illustration purposes only, note that the su-perframe period in Table 3 is deliberately chosen to be different for the two channels.
Table 3
Superframe table for the FDD-RR protocol.
t1 t2 · · · t`y t`y+1 CH1 P → D y 1 D y 1 → D y 2 · · · D y `y−1 → D y `y D y `y → C CH2 C → Du1 Du1 → Du2 · · · Du`u → P
In this case, we have that δy α(i) . = 1 − 1Dyα(i), δ u β(i) . = 1 − 1Du β(i), γ y α(i) . = 1 − 1D¯y α(i), and γ u β(i) . = 1 − 1D¯u β(i), where Dαy . = {i ∈ N : i = α + (`y+ 1)σ, σ ∈ Z≥0}, Dβu . = {i ∈ N : i = β + (`u+ 1)σ, σ ∈ Z≥0}, ¯ Dy α . = {i ∈ N : i = α + 1 + (`y+ 1)σ, σ ∈ Z≥0}, ¯ Dyβ= {i ∈ N : i = β + 1 + (`. u+ 1)σ, σ ∈ Z≥0}
for α = 1, . . . , `y and β = 1, . . . , `u. For this case, the
parameter T in Lemma 8 is given by T = TFDD-RR . = max{2`y+ 1, 2`u+ 1}.
5.3.3 Wave Round Robin (W-RR)
This scheduling protocol schedules devices in an inter-leaved manner, see Table 4. That is, device one receives the measurement from the plant in the first timeslot, and at the same exact time (but in different frequency channels), device two transmits to device three and so on. Note that the number of channels used for the
y-Table 4
Superframe table for the W-RR protocol (`yeven, `uodd).
t1 t2 CH1 P → Dy1 D y 1 → D y 2 CH2 D y 2 → D y 3 D y 3 → D y 4 .. . ... ... CHMy−1 Dy`y−2→ D y `y−1 D y `y−1 → D y `y CHMy D y `y → C CHMy+1 C → D u 1 Du1 → Du2 CHMy+2 D2u→ D3u Du3 → Du4 .. . ... ...
CHMy+Mu−1 D`uu−3→ D`u−2u Du`u−2→ Du`u−1
CHMy+Mu D u
`u−1→ Du`u D u `u → P
the u-path (namely Mu) satisfy My . = `y+2−θy 2 , Mu . = `u+2−θu
2 , where θy, θu ∈ {0, 1}. In particular, θy (resp.
θu) is 0 if `y (resp. `u) is even, and 1 if `y (resp. `u)
is odd. Furthermore, it is important to note that in WH networks, the available number of channels is fixed (equal to 15). Therefore, My and Mu need to satisfy
My+ Mu≤ 15. However, this is not a problem because
if there are too many field devices, the superframe pe-riod can always be increased to schedule the remaining devices that did not fit the 15 channels into another W-RR starting immediately after timeslot two.
For this example, δ1+2α1y (i) = 1 − 1. W(i), δy2+2α2(i)
. = 1W(i), δ1+2βu 1(i) . = 1 − 1W(i), δu2+2β2(i) . = 1W(i), γy α(i) . = 1−δy
α(i), and γβu(i)
. = 1−δu β(i), where W . = {i ∈ N : i = 1 + 2σ, σ ∈ Z≥0}, for α1 = 0, 1, . . . ,`y+θy2 −2, α2 = 0, 1, . . . , `y−θy−2 2 , β1 = 0, 1, . . . , `u+θu−2 2 , β2= 0, 1, . . . ,`u−θu−22 , α = 1, . . . , `y, and β = 1, . . . , `u.
For this case the parameter T in Lemma 8 is given by T = TW-RR
.
= max{`y+ 2, `u+ 2}.
Remark 10 It can be seen that, exploiting multiple fquencies channels reduces the amount of timeslots re-quired in the superframe (cf. Tables 2, 3 and 4). Specif-ically, it reduces the parameter T related to the P ET
property of the protocol. We note that, in particular, TS-RR> TFDD-RR≥ TW-RRfor all `y, `u6= 0. This will
be-come important in Sections 6 and 8 when comparing the MATI bounds of these three protocols. 6 Lpstability of WH–NCSs
In this section, we prove that P ET protocols lead to Lp
stability of the WH–NCS. Inspired by the results in [24], we show that by exploiting the structure of our model, we can further improve the MATI bounds in [24]. Note that g in (12) has a particular structure in which several rows are equal to zero. To better reveal this structure from our model (10), we re-arrange the error vector ζ = (ζy, ζu)
in (9) via a change of coordinates. That is, we define ζ= T ζ, where T is a matrix such that.
˙ ζ = T g(x, ζ, w)= g(x, ζ, w). = −∂gp ∂xpfp(xp, I 2`u nu · ζu+ gc(xc), w) −∂gc ∂xcfc(xc, I 2`y ny · ζ y+ g p(xp), w) 0 .. . 0 (15)
Note that the first two components of ˙ζ are non-zero while the rest are zero. This mathematical structure is key in the following stability results, and it comes di-rectly from our modelling study in Section 4.
We impose the following assumption on the ζ-subsystem.
Assumption 11 Let L11 ∈ A nζy 1 +nζu 1 ≥0 and L12 . = L11diag{I 2`y−1
ny , I2`nuu−1}. There exists a matrix A ∈ A nζ ≥0 of the form A =L11 L12 0 0 , (16)
and a continuous function ˜y : Rnx × Rnw → Rnζ + such
that the error dynamics (10b) satisfy1
¯˙ζ = ¯g(x, ζ, w) A¯ζ + ˜y(x, w), (17) for all (x, ζ, w) ∈ Rnx× Rnζ× Rnw.
Assumption 11 is the vector analogue of the dissipation type inequality imposed on the network-induced error system in [18], or many other works on non-linear NCSs [17, 19, 27]. This type of inequality essentially requires that the network-induced error exponentially grows dur-ing flows. Such a property is natural, as the ζ-system is typically unstable between two transmission instants.
Assumption 11 is also particularly inspired by (38) in [24]. However, in [24], the function g does not have the structure in (15) that follows directly from our model. Note that ¯g has the form (¯g1, ¯g2, 0, · · · , 0), and that g1
and g2depend on sums of the components of ζ (instead
of the whole error as in [24]). Then, we can assume a lin-ear bound on each component as in [24], i.e. we assume that there exist L1 ∈ A
ny×nu ≥0 , L2 ∈ A nu×ny ≥0 and func-tions ˜y1: Rnx× Rnw → Rny and ˜y2: Rnx× Rnw→ Rny, such that ¯g1(x, I2`nuu · ζ u, w) ≤ L 1( ¯ζ1u+ · · · + ¯ζ u 2`u) + ˜ y1(x, w), ¯g2(x, I 2`y ny ·ζ y, w) ≤ L 2( ¯ζ y 1+· · ·+ ¯ζ y 2`y)+˜y2(x, w).
Then, given the definition of ζ, we naturally get the bound in (17), i.e. ¯ g 0 L1 0 L1I2`u−1nu L2 0 L2I 2`y−1 ny 0 0 0 0 0 .. . ... ... ... 0 0 0 0 ¯ ζ + ˜ y1(x, w) ˜ y2(x, w) 0 .. . 0 ,
for ? ∈ {y, u}, and thus L11, L12and ˜y(x, w) in
Assump-tion 11 follow. We will show in SecAssump-tion 7 that the block structure of A in (16) arises naturally in linear systems.
This next theorem states, in essence, that for sufficiently small MATI, the class of P ET protocols in Section 5.2
lead to the finite Lpstability of the ζ-subsystem, which
is key to prove Theorem 4 below. The proof can be found in the appendix.
1
Theorem 3 Suppose that the scheduling protocol (10d) is persistently exciting in time T and that Assumption 11 holds. Further suppose that MATI satisfies τ ∈ [ε, τ∗),
ε ∈ (0, τ∗) where τ∗ = ln 1 + 1/√% (|L11|T ), and
%= max{2`. y, 2`u}. Then, the system (10b), (10d) is Lp
stable from ˜y to ζ for p ∈ [1, ∞] with gain
˜
γ(τ ) = T exp(|A|(T + 1)τ )(exp(|A|τ ) − 1) |A| 1 −√% (exp(|L11|T τ ) − 1)
. (18) The following theorem provides the closed-loop Lp
sta-bility result for the WH–NCS in (10), and is the main re-sult of this section. It asserts that P ET protocols lead to
Lpstability of the WH–NCS for sufficiently small MATI.
Theorem 4 Consider the WH–NCS (10) and suppose that
(1) The conditions of Theorem 3 hold with ˜y = G(x) + w;
(2) (10a) is Lp stable from (ζ, w) to G(x) with gain γ
for some p ∈ [1, ∞];
(3) MATI satisfies τ ∈ [ε, τ∗), ε ∈ (0, τ∗), where τ∗ = ln(z)/(|A|T ), A comes from Assumption 11, and z solves
γT z1+2/T − γT z1+1/T +√%|A|z|L11|/|A|
− (1 +√%) |A| = 0. (19) Then, the WH–NCS is Lpstable from w to (G(x), ζ) with
linear gain.
PROOF. The proof follows from the fact that ˜γ(τ ) in (18) is differentiable and monotonically increasing in τ for τ ∈ [ε, ln(1 + 1/√%)/|L11|T ] and thus, in view of
the inverse function theorem [22], there exists a unique solution τ∗ to ˜γ(τ )γ = 1. That is, γT exp(|A|(T + 1)τ )(exp(|A|τ )−1)−|A| 1 −√% (exp(|L11|T τ ) − 1) =
0, for which we define z= exp(|A|T τ ) and thus get (19).. Then, by monotonicity of ˜γ(τ ), we have that ˜γ(τ )γ < 1 for any τ ∈ [ε, τ∗). The proof is complete in light of the
small-gain Theorem 1.
It is not immediately clear how the MATI bound τ∗ behaves for different values of T . From Remark 10 we know that TS-RR > TFDD-RR ≥ TW-RR for all `y, `u 6=
0. Intuition suggests that protocols exploiting multiple frequencies to schedule devices, i.e. smaller T , would lead to larger MATI bounds. In fact, if we take less time to get a packet from plant to controller (or vice-versa) then the inter-transmission bound may not need to be small in order to preserve stability. However, if we design a scheduling protocol in which the packet takes longer to reach the controller (e.g. S-RR), then faster transmission
is required in order to preserve stability. Hence, we would expect that τS-RR∗ ≤ τ∗
FDD-RR ≤ τ
∗
W-RR. In Section 8, we provide numerical examples in which this statement indeed holds.
In the proof of Theorem 4, the Lp stability property is
obtained from an L1bound, see (A.19). Such L1 bound
is computed under the definition of k · kLp in (1) with
k · k= | · |. However, using k · k. = k · k. p in (1) to define
k · kLp(and thus k · kL1) will be particularly useful given
the structure of A. In particular, we will see that this approach provides larger MATI bounds than the ones obtained in Theorem 4. To be precise and consistent with the above discussion, in Theorem 5 and Theorem 6, the definition of Lpstability is as per Definition 1 with k·kLp
as in (1), with k · k= k · k. p.
Theorem 5 Suppose that the scheduling protocol (10d) is persistently exciting in time T and that Assumption 11 holds. Further suppose that MATI satisfies τ ∈ [ε, τ∗), ε ∈ (0, τ∗) where τ∗= ln(2)(kL11k1T ). Then, the NCS
error subsystem (10b)-(10d) is Lpstable from ˜y to ζ for
p ∈ [1, ∞] with gain
˜
γ(τ ) = T exp(kL11k1(T + 1)τ )(exp(kL11k1τ ) − 1) kL11k1(2 − exp(kL11k1T τ ))
.
PROOF. The proof follows along similar lines to the proof of Theorem 3. However, we use the definition of k · kLp as in (1) with k · k
.
= k · kp. Specifically, we use
this definition in (A.19) when computing the L1bound.
Therefore, instead of having inequalities with Euclidean norm as in (A.18), we now bound k ¯ζ(ϑ)k1so that we get
k ¯ζkL1[tk,tk+1] with the current definition. We point out
here the important differences, in which the structure of A plays a crucial role. The first significant difference appears when dealing with terms of the form (cf. (A.4))
exp(AT τ ) − I =exp(L11T τ ) − I 0 0 0 I Iy,u 0 0 , where Iy,u .
= diag{I2`yny−1, I2`u−1nu }. By definition of the
induced 1-norm of a matrix, definition of matrix expo-nential and triangle inequality, we have that
exp(AT τ ) − I 1 ≤ exp(L11T τ ) − I 0 0 0 1 I Iy,u 0 0 1 = kexp(L11T τ ) − Ik1· 1 ≤ exp(kL11k1T τ ) − 1. (20)
The other significant difference is that we can bound terms of the form exp(At), for any t ≥ 0, as follows k exp(At)k1= k exp(At) − I + Ik1≤ k exp(At) − Ik1+
kIk1. Now, we can use (20) in the last inequality to
in-duced 1-norm suits the structure of A perfectly in the sense that we can now bound the exponential matrix exp(At) by a matrix that only depends on L11.
Theorem 6 Consider the WH–NCS (10) and suppose that
(1) The conditions of Theorem 5 hold with ˜y = G(x) + w;
(2) (10a) is Lp stable from (ζ, w) to G(x) with gain γ
for some p ∈ [1, ∞];
(3) MATI satisfies τ ∈ [ε, τ∗), ε ∈ (0, τ∗), where τ∗ = ln(z)/(kL11k1T ) and z solves
γT z1+2/T− γT z1+1/T + z kL
11k1− 2kL11k1= 0.
Then, the WH–NCS is Lpstable from w to (G(x), ζ) with
linear gain.
7 Case study: The linear case
Consider the WH–NCS of Fig. 4 where the plant and controller have the following linear state-space form
˙
xp = Apxp+ Bpu,ˆ x˙c= Acxc+ Bcy,ˆ
y = Cpxp, u = Ccxc.
(21)
Recall the network-induced error in (9) and the aug-mented state x = (xp, xc), then (10a)-(10b), together
with the re-arranged error ζ, becomes
˙ x = A11x + A12ζ (22a) ˙ ζ = A21x + A22ζ. (22b) where A11 . = Ap BpCc BcCp Ac , A12 . = 0 Bp 0 BpI 2`y−1 ny Bc 0 BcI2`nuu−1 0 , A21 . = −[Cp 0]A11 −[0 Cc]A11 0 .. . 0 , A22 . = 0 −CpBp 0 −CpBpI2`u−1nu −CcBc 0 −CcBcI 2`y−1 ny 0 0 0 0 0 .. . ... ... ... 0 0 0 0 .
With the above, we can state the following immediate result.
Proposition 1 Consider the linear system (22). Then, Assumption 11 is satisfied with A = A22, and ˜y(x) =.
A21x, for any x ∈ Rnx, and thus Theorems 4 and 6 can be
directly invoked to compute the MATI bound that ensures Lp stability of the linear system (22).
Note that Assumption 11 is naturally satisfied in linear WH–NCSs, given the structure of the WH network.
8 Numerical examples
8.1 MATI Computation
In this section, we numerically compare our MATI bounds to the bounds in [17, 24] on an example. We restrict the discussion of this section to linear time-invariant systems in the absence of exogenous distur-bances, and we thus verify UGES via Theorem 2.
Table 5
MATI bounds achieving UGES for the linear example in Section 8.1, when the three P ET scheduling protocols of
Section 5.3 are implemented. (`y= 2, `u= 1.)
S-RR FDD-RR W-RR τ∗[17]in [ms] 5.11 · 10−5 4.18 · 10−3 3.86 · 10−2 τ∗[24]in [ms] 13.28 18.37 22.72 τthm.4∗ in [ms] 13.53 18.69 23.1 τthm.6∗ in [ms] 17.54 24.31 30.11 τ∗ thm.4 vs. τ∗[17] 2.65 · 10 7% 4.47 · 105% 5.97 · 104% τthm.4∗ vs. τ∗[24] 1.82% 1.75% 1.69% τthm.6∗ vs. τ∗[17] 3.43 · 107% 5.81 · 105% 7.79 · 104% τthm.6∗ vs. τ∗[24] 32% 32.3% 32.55%
Consider the linear WH–NCS in (22) with Ap =
0.5, Bp= 1.1, Cp= 1, Dp= 0, Ac= −2, Bc = −1, Cc =
1.5, Dc= 0, and `y= 2, `u= 1. In this case,
A = 0 1.1 0 0 0 1.1 1.5 0 1.5 1.5 1.5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ,
and we note that L11=
0 1.1 1.5 0
, |A| = 3, and |L11| =
kL11k1 = 1.5. We implement the three scheduling
pro-tocols of Section 5.3, for which TS-RR= 7, TFDD-RR = 5 and TW-RR = 4. We can compute the L2 gain from ζ
to ˜y = A21x using Matlab and have γ = 5.19. With
the above, Table 5 is constructed by using Theorems 4 and 6 to compute the MATI bounds τthm.4∗ and τthm.6∗ (both theorems can be directly invoked given Proposi-tion 1). The bound τ∗
[17]is computed by using our
log10(ℓy) 0 0.5 1 1.5 lo g10 (% of in cr em en t) -0.5 0 0.5 1 1.5 τthm.3∗ vs τ[19]∗ τ∗ thm.4vs τ[19]∗ log10(ℓy) 0 0.5 1 1.5 lo g10 (% of in cr em en t) 3.9 3.92 3.94 3.96 3.98 4 τthm.3∗ vs τ∗ [13] τ∗ thm.4vs τ[13]∗
Fig. 5. Linear example: Percentage of improvement between τthm.4∗ , τ
∗
thm.6and τ ∗
[24](left), and between τ ∗ thm.4, τ
∗ thm.6and
τ∗[17] (right). (`y∈ [1, 20] and W-RR.)
framework in [24], which does not exploit the structure of A. Next, we obtain the bounds τthm.3∗ and τthm.4∗ , in view of Theorems 4 and 6. Note that we can use the same L2gain γ in both Theorems 4 and 6 given that k · kLpin
(1) with k · k= | · |, for p = 2, coincides with k · k. Lpwhen
using k · k= k · k. p. In order to have a clear comparison,
we have also included, in the last four rows, the per-centage of improvement between our proposed bounds τthm.4∗ , τthm.6∗ and previous literature τ∗[17], τ∗[24]. (We de-fine the percentage of improvement between τA> 0 and
τB> 0, where τA≥ τB, as 100 × (τA− τB)/τB.)
We make the following comments:
(1) We can see that implementing a W-RR protocol results in larger MATI bounds in comparison to S-RR and FDD-S-RR. This is consistent with intuition. (2) Our MATI bounds τthm.4∗ and τthm.6∗ are both larger than the bounds in previous works [17] and [24], given we exploit the mathematical structure of our WH–NCS model. In fact, the bound τ∗[17] is quite conservative in the context of WH.
(3) The bound τ∗[17]= 3.86 · 10−2[ms] (W-RR column) that preserves UGES is equivalent to a network throughput of 26.32 Mbps (WH has a maximum packet length of 127 bytes). However, the maxi-mum data rate allowed in WH networks is 250 kbps, meaning that the bounds in [17] are quite conser-vative. Our current bounds τthm.3∗ = 23.1[ms] and τthm.4∗ = 30.11[ms], are equivalent to 43.98 kpbs and 33.74 kbps, respectively. These are actually achievable on current WH networks.
(4) When using Theorem 4, the corresponding bounds are larger by around 1.7%. This is a modest im-provement but still tighter than [24]. Theorem 6 provides MATI bounds around 33% better. In addi-tion, Figure 5 shows that the percentage of improve-ment between our bounds and [17,24] increases with the number of field devices.
0 2 4 6 8 10 Time (seconds) 0 2 4 6
Without clock drift With clock drift
Fig. 6. Stabilisation of the plant (21) when the controller is implemented over a WH network via the TrueTime sim-ulator without clock drift (solid line) and with clock drift (dashed line).
8.2 TrueTime simulation of the WH–NCS
We now present a simulation in TrueTime, which is a Simulink-based simulator for real-time control sys-tems, including controller task execution, real-time ker-nels, network transmissions and continuous plant dy-namics [12]. Consider the same linear plant and con-troller in Section 8.1. We have implemented the FDD-RR protocol from Section 5.3.2, in which the sensor mea-surements are sent in a RR fashion via channel one, and the control signal is sent via channel two. In the y-path we one sensor and one router (i.e. `y = 2), and in the
u-path we have one actuator (i.e. `u= 1). All the
commu-nications are done through a WH network under TDMA communications. The controller and every field device are simulated with TrueTime kernel blocks, which are responsible for control logic, network data acquisition, data processing and calculations. The TrueTime net-work block simulates the transfer of packets in a real network, in this case, a WH network.
In Figure 6, we can see that the controller can suc-cessfully stabilise the plant from an initial condition of xp(0) = 5 when implemented over a WH network. In
the same figure, we supposed that the actuator is sub-ject to clock drift with respect to the field devices in the y-path, which can be explicitly added in the True-Time kernel block. That is, we are perturbing Assump-tion 4(a), and thus transmissions in channel one might not happen at the same time as transmissions in channel two. The usual clock drift in digital devices is around 10 parts-per-million (ppm), and we have used this value in the simulation. We can see from Figure 6 that the con-troller is still capable of stabilising the plant even when field devices are subject to clock drift.
9 Conclusions
This paper studied general non-linear control systems with disturbances that are implemented over WH net-works. We first provided a hybrid model of WH–NCSs, which captures most functionalities of the network. Such model is then used to study Lpstability of WH–NCSs. In
rates, a scheduling protocol that regularly schedules to transmit every field device within a fixed period of time ought to preserve Lpstability of the network-free system.
We thus provided guidelines to design scheduling proto-cols, implementable in WH under TDMA communica-tions, such that this property is indeed satisfied. Quanti-tatively, this paper provided sharp MATI bounds, signif-icantly improving upon bounds provided in [17] and [24]. This improvement relies exclusively on exploiting the mathematical structure of our model. Future work will focus on studying the tracking problem, and also on ob-taining results for more general network topologies.
A Appendix
A.1 Technical lemmas
In order to prove the main results of this paper, we first need the following lemmas.
Lemma 12 Let A, B ∈ An
≥0, and suppose A B. Then:
∀x ∈ Rn
≥0, Ax Bx; ∀C ∈ An≥0, AC BC; and ∀C ∈
An
≥0, CA CB.
Lemma 13 (cf. [24, 25]) Let v ∈ Rn and con-sider Dv(t) Av(t) + d(t) with v(t0) = v0, for
all t ∈ I ⊂ R, where Dv(t) denotes the left-handed derivative of v(t), A ∈ An
≥0 and d(t) : I → Rn
is continuous. Then, v(t) is bounded by v(t) exp(A(t−t0))v0+
Rt
t0exp(A(t−s))d(s) ds, for all t ∈ I.
Lemma 14 Suppose that A ∈ An≥0 has the block struc-ture in (16), and {Qi}i∈N∈ Sn(T ) are arbitrary. Then
Q
n+T −1
i=n Qiexp(Aτ )
≤ λ < 1, for all n ∈ N, and all τ ∈ [0, τ∗), where τ∗= ln 1 + 1/√% (|L11|T ), and
λ= (exp(|L. 11|T τ ) − 1)
√
%, and %= max{2`. y, 2`u}.
PROOF. Fix n ∈ N and τ ∈ R. By following the in-duction procedure in the proof of Lemma 7.1 in [24], to-gether with properties (2) and (3) of Lemma 12, we get
n+T −1 Y i=n Qiexp(Aτ ) ! n+T −1 Y i=n Qi ! + exp(AT τ ) − I.
Then, given that {Qi}i∈N∈ Sn(T ),
n+T −1 Y i=n Qiexp(Aτ ) !
≤ | exp(AT τ ) − I|. (A.1)
We now exploit the structure of A. Note that since A has the block structure in (16), we have that, by definition of the matrix exponential
exp(AT τ ) − I =L11L12 0 0 T τ + 1 2! L2 11 L11L12 0 0 (T τ )2+ · · · (A.2)
Now, note that by definition of L12 in Assumption 11,
we have that L12T τ + 1 2!L11L12(T τ ) 2 + · · ·
= (exp(L11T τ ) − I)Iy,u. (A.3)
Given (A.2) and (A.3), we have that
exp(AT τ ) − I =exp(L11T τ ) − I 0 0 0 I Iy,u 0 0 . (A.4)
Therefore, the induced 2-norm of exp(AT τ ) − I can be bounded as follows | exp(AT τ ) − I| ≤ exp(L11T τ ) − I 0 0 0 I Iy,u 0 0 = | exp(L11T τ ) − I| p max{%}, (A.5)
where the last equality follows from the definition of 2-norm and Iy,u. Then,
| exp(L11T τ ) − I| ≤ |L11|T τ +
1 2!|L11|
2(T τ )2+ · · ·
= exp(|L11|T τ ) − 1. (A.6)
Putting together (A.1), (A.5) and (A.6) we can conclude
n+T −1 Y i=n Qiexp(Aτ ) ! ≤ | exp(AT τ ) − I| ≤ (exp(|L11|T τ ) − 1) √ %. (A.7)
It is clear now that we need to choose τ such that (A.7) is less than one. We thus let (exp(|L11|T τ∗) − 1)
√ % = 1, and get τ∗= ln 1 + 1/√% /(|L11|T ). By monotonicity
of (exp(|L11|T τ ) − 1)
√
% in τ , (A.7) is less than one for all [0, τ∗), completing the proof.
A.2 Proof of Lemma 8
Consider a scheduling protocol that satisfies Assumption 7. Then, the corresponding protocol equation is given by (10d) with H(i) satisfying (13). Now, consider the following auxiliary discrete-time system
ζ(k + 1) = H(k)ζ(k), (A.8)
initialized at time i with initial condition ζ(i) = ζ. The solution of this system at time k, starting at time i and
initial condition ζ is denoted by φ(k, i, ζ), and it is given by φ(k, i, ζ) = Qk−1
j=i H(j)ζ. Now, recall from
Sec-tion 5.1 that H(i) depends on γαy(i), γβu(i), δαy(i), δβu(i) which take values in {0, 1}, and these are related to the components of ζ being reset to zero whenever transmis-sions take place. Therefore, if every field device transmits within T units of time, it means that every component of the error would have been reset to zero in that period. Therefore, there exists T ∈ N such that the solution of the discrete-time system (A.8) satisfies φ(T + i, i, ζ) = 0 for all i ∈ N and ζ ∈ Rnζ. That is, QT +i−1
j=i H(j)ζ = 0,
for all i ∈ N and ζ ∈ Rnζ. Because the latter holds for
every ζ ∈ Rnζ, we conclude that H satisfies (14) for some
T ∈ N, completing the proof.
A.3 Proof of Theorem 3
Define Qi
.
= H(i) for each i ∈ N, and assume the sys-tem is initialised at time ts, ts ∈ [0, t0], t0− ts < τ .
Given that the protocol is P ET, then {Qi}i∈N∈ Snζ(T ).
For simplicity, we write ˜y(s) instead of ˜y(x(s), w(s)). By hypothesis, we have that ¯g(x, ζ, w) = ¯˙ζ A¯ζ + ˜y(t), for almost all t ∈ [ti, ti+1]. The ι-th component of ¯˙ζ,
ι ∈ {1, . . . , 2`y+ 2`u}, is given by d dtζι(t) = lim h→0,h<0 ζι(t + h) − ζι(t) h ≥ lim h→0,h<0 |ζι(t + h)| − |ζι(t)| h = D ¯ζι(t), hence D ¯ζ A ¯ζ + ˜y(t). We now apply Lemma 13 to this equation with initial condition ¯ζ(t+i−1), together with the protocol equation (10d), to get
¯
ζ(t+i ) Qiexp(A(ti− ti−1)) ¯ζ(t+i−1)
+ Qi
Z ti
ti−1
exp(A(ti− s))˜y(s)ds. (A.9)
Define Ri
.
= Qiexp(Aτ ). By iterating the linear
recur-rence (A.9) from the initial condition ¯ζ(ts), and using
Corollary 1, we get the following bound for all k ∈ Z≥0
¯ ζ(t+k) k Y i=0 Ri ! ¯ ζ(ts) + k X i=0 k Y n=i Rn ! × exp(−Aτ ) Z ti ti−1
exp(A(ti− s))˜y(s)ds. (A.10)
Fix τ ∈ (0, τ∗), where τ∗ comes from Lemma 14. We first set the disturbance term ˜y = 0 and we focus in the contribution of the initial condition ¯ζ(ts). Then, by
using Lemma 14, we have that, for all m ∈ N | ¯ζ(t+mT −1)| ≤ mT −1 Y i=0 Ri | ¯ζ(ts)| ≤ λm| ¯ζ(ts)|. (A.11)
From D ¯ζ A ¯ζ + ˜y(t) with ˜y = 0, we have that ¯ζ(s) exp(A(s − s0)) ¯ζ(s0) (cf. Lemma 13), for every ¯ζ(s0) ∈
Rnζ. From the latter, and by using ¯ζ(s
0) = ¯ζ(t+mT −1)
and its bound in (A.11), we have that
| ¯ζ(θ)| ≤ exp(|A|(θ − tmT −1))λm| ¯ζ(ts)|, (A.12)
for all m ∈ N and θ ∈ (tmT −1, t(m+1)T −1).
Rais-ing (A.12) to the p-th power and integratRais-ing over [tmT −1, t(m+1)T −1], we get ¯ζ p Lp[tmT −1,t(m+1)T −1]≤ λmp p|A|(exp(|A|pT τ ) − 1)| ¯ζ(ts)| p , (A.13)
for all m ∈ N and all p ∈ [1, ∞). Now, we need to com-pute a similar bound in the interval [ts, tT −1].
Proceed-ing as above, we have that
¯ζ p Lp[ts,tT −1]≤ 1 p|A|(exp(|A|pT τ ) − 1)| ¯ζ(ts)| p, (A.14)
for all p ∈ [1, ∞). Therefore, by summing (A.14) and (A.13) with m → ∞, and taking the p-th root, we have that, for all t ≥ ts
¯ζ Lp[ts,t]≤ ∞ X i=0 λi exp(|A|pT τ ) − 1 p|A| 1/p | ¯ζ(ts)| ≤ 1 1 − λ exp(|A|pT τ ) − 1 p|A| 1/p | ¯ζ(ts)|, (A.15)
where the last inequality follows from λ < 1 given Lemma 14. For p = ∞ we get the L∞bound by taking
limp→∞
¯ζ
L
p[ts,t] in (A.15), that is,
¯ζ L∞ ≤ 1 1 − λexp(|A|T τ )| ¯ζ(ts)|. (A.16) We now focus on bounding the contribution from the disturbance term in (A.10), obtained by setting ¯ζ(ts) =
0. We get ¯ ζ(t+k) k X i=0 k Y n=i Rn ! exp(−Aτ ) × Z ti ti−1
Again, applying Lemma 13 to D ¯ζ A ¯ζ + ˜y(t) with initial condition (A.17), we get, for ϑ ∈ [tk, tk+1]
¯ ζ(ϑ) exp(A(ϑ − tk)) k X i=0 k Y n=i Rn ! × exp(−Aτ ) Z ti ti−1 exp(A(ti− s))˜y(s)ds + Z ϑ tk exp(A(ϑ − s))˜y(s)ds.
We now use the same division algorithm used in [24, Theorem 5.1] to show that
k Y n=i Rn ≤ λb(k+1−i)/T cexp(|A|(T − 1)τ ).
With the above we can get the following bound for all ϑ ∈ [tk, tk+1] | ¯ζ(ϑ)| ≤ exp(|A|(ϑ − tk)) exp(|A|T τ ) k X i=0 λb(k+1−i)/T c × Z ti ti−1 exp(|A|(ti− s))|˜y(s)|ds + Z ϑ tk
exp(|A|(ϑ − s))|˜y(s)|ds. (A.18)
In order to compute the Lp bound in this case, we will
start by finding an L1estimate. Define ϕ(s)
.
= exp(|A|s). We integrate (A.18) and use Young’s inequality [9] to get
¯ζ L1[tk,tk+1]≤ kϕkL1[0,τ ]exp(|A|T τ ) × k X i=0 λb(k+1−i)/T c Z ti ti−1 exp(|A|(ti− s))|˜y(s)|ds + kϕkL1[0,τ ]k˜ykL1[t k,tk+1].
We can further bound the above as
¯ζ L1[tk,tk+1]≤ kϕkL1[0,τ ]exp(|A|(T + 1)τ ) × k X i=0 λb(k+1−i)/T ck˜ykL1[ti−1,ti] + kϕkL1[0,τ ]k˜ykL1[tk,tk+1] ≤ kϕkL1[0,τ ]exp(|A|(T + 1)τ ) × k+1 X i=0 λb(k+1−i)/T ck˜ykL1[t i−1,ti]. (A.19)
As in [24, Theorem 5.1], the bound on k ¯ζkL∞[tk,tk+1] is
similar and omitted for brevity. Then, we can use the Riesz-Thorin interpolation theorem [23, p.52] to bound the Lpnorm by the above L1estimate. That is, summing
(A.19) over the interval [ts, tM], for all M ≥ 0, we get
¯ζ Lp[ts,tM]≤ kϕkL1[0,τ ]exp(|A|(T + 1)τ ) × M −1 X k=−1 k+1 X i=0 λb(k+1−i)/T ck˜ykLp[ti−1,ti],
for p ∈ [1, ∞]. We now apply the Discrete Young’s In-equality [24, Lemma 1.1], and take the limit as M → ∞ in the above summation to get
¯ζ Lp[ts,tM] ≤ kϕkL1[0,τ ]exp(|A|(T + 1)τ ) × ∞ X i=0 λbi/T c ∞ X i=0 k˜ykLp[t i−1,ti] = kϕkL1[0,τ ]exp(|A|(T + 1)τ ) × T 1 − λ k˜ykLp[ts,tM] =T exp(|A|(T + 1)τ )(exp(|A|τ ) − 1) |A| 1 +√% −√% exp(|L11|T τ ) k˜ykLp[ts,tM], (A.20)
where the last equality comes from the definition of ϕ, and the definition of λ in Lemma 14. Note that either k˜ykLp[ts,tM] = 0 or the ratio ¯ζ
L
p[ts,tM]/ k˜ykLp[ts,tM]
is bounded by an expression that is independent of M , hence, (A.20) remains true with t in lieu of tM for any
t ≥ ts. To finish the proof, we sum both (A.15) and
(A.20), to get ¯ζ L p[ts,t]≤ 1 1 −√% (exp(|L11|T τ ) − 1) × exp(|A|pT τ ) − 1 p|A| 1/p ¯ζ(ts) +T exp(|A|(T + 1)τ )(exp(|A|τ ) − 1) |A| 1 −√% (exp(|L11|T τ ) − 1) k˜ykLp[ts,t], (A.21)
for p ∈ [1, ∞). The L∞ bound is obtained by summing
(A.16) and (A.20), that is
¯ζ L∞[ts,t] ≤ exp(|A|T τ ) 1 −√% (exp(|L11|T τ ) − 1) | ¯ζ(ts)|+ T exp(|A|(T + 1)τ )(exp(|A|τ ) − 1) |A| 1 −√% (exp(|L11|T τ ) − 1) k˜ykL∞[ts,t]. (A.22)
![Fig. 5. Linear example: Percentage of improvement between τ thm.4∗ , τ thm.6∗ and τ ∗ [24] (left), and between τ thm.4∗ , τ thm.6∗ and τ ∗ [17] (right)](https://thumb-eu.123doks.com/thumbv2/123doknet/14449078.518297/13.892.484.815.114.251/fig-linear-example-percentage-improvement-thm-left-right.webp)
